RADIANS Definition An arc of length r subtends an angle of one radian at the centre of a circle of radius r r Ø=1radian.

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Presentation transcript:

RADIANS Definition An arc of length r subtends an angle of one radian at the centre of a circle of radius r r Ø=1radian

Proof r How do you calculate the length of an arc? r = ø x 2πr 360º Ø=1radian How do you calculate the length of an arc? r = ø x 2πr 360º r = 1 radian x 2πr 360 º x 360 º 360r = 1 radian x 2πr ÷ r 360 º = 1 radian x 2π ÷ 2 180 º = 1 radian x π Or 180 º = π radians ÷ π 180 = 1 radian π so 1 radian is approximately…?

Converting between angles and radians Degrees = radians x 180 π Radians = degrees x π 180 So if ø is measured in radians Then ø radians = ø x π 180

How many different angles can you write as radians? 2 180º = π radians

Arc Length r Angle in degrees Arc length = ø x 2πr 360º Ø=1radian Angle in degrees Arc length = ø x 2πr 360º Arc length = 2πrø 360º Factorise r Arc length = r 2πø 360º Divide by 2 Angle in radians Arc length = r πø 180º Arc length = rø

Area of Sector r Angle in degrees Sector area = ø x πr2 360º Ø=1radian Angle in degrees Sector area = ø x πr2 360º Sector area = πr2ø 360º Factorise r2 Sector area = r2 πø 360º Factorise out ½ Angle in radians Sector area = ½r2 πø 180º Sector area = ½r2ø

Examples Convert 50° into radians 50° = 50° x π rad 180 50° = 0.87 rad

Examples Convert 2.7 radians into degrees 2.7 rad = 2.7 x 180 degrees π 2.7 rad = 154.7 °

Examples Convert 40° into radians 40° = 40 x π 180 40° = 40π 180 40° = 40π 180 40° = 2π radians 9

Calculate the arc length and sector area 10cm 1.2radians Arc length = rө Arc length = 10 x 1.2 Arc length = 12cm Sector area = ½r2ө Area = ½ x 100 x 1.2 Area = 60cm2